E624 Mathematical Economics I
"Stochastic Optimization in Continuous Time"
by Professor Fwu-Ranq Chang
This is an introductory course on stochastic control theory with
applications to economics. The focal point of the mathematical
technique is the stochastic dynamic programming method. It includes
the essential elements of Ito's calculus (Wiener process, stochastic
integrations and Ito's lemma), the stochastic differential equations
(existence, uniqueness, and some closed-form solutions), and the
stochastic Bellman equations. These mathematical results will be
integrated with economic problems. The emphasis of the course is on
problem solving, not on proving general theorems. One of the
objectives is that you will be able to apply the Bellman equation to
dynamic economic problems the same way you apply the Lagrange
Multiplier method to static economic problems.
Applications to economics are many and selective. We will cover the
traditional optimal growth theory, including the inverse optimal
method. We will also cover adjustment cost theory of supply,
irreversible investment, exhaustible resources, optimal consumption
and portfolio rules, index bonds, uncertain lifetimes and life
insurance, and, naturally, the one and only Black-Scholes option
pricing theory. Another class of economic applications will be covered
in this course is the "barrier" problem. We shall build our absorbing
barrier theory based on the famous Baumol-Tobin transactions demand
for money model.
The "text" for the course is the manuscripts that I have prepared. It
is part of a book project, which so far has six chapters (about 200
pages) typeset in Scientific Word 2.5. They are:
Chapter one: Probability Theory
Chapter two: Wiener Process
Chapter three: Stochastic Calculus
Chapter four: Stochastic Dynamic Programming
Chapter five: How to Solve It
Chapter six: Barrier and Uncertain Lifetimes
There will be a midterm exam and the final exam.